The state of an electron in a hydrogenic atom is given by the un-normalised wavefunction

Φ = \(\left\{Y_{10}(\theta, \phi) +\frac{1}{\sqrt2}Y_{11}(\theta, \phi)\right\}R(r)\)

where Y_im (θ, ϕ) are spherical harmonics and R(r) is the radial function. The probability that a measurement of L_z will give an eigenvalue of h is

Concept:

The hydrogenic wavefunctions are solutions to the Schrödinger equation for the hydrogen atom (or hydrogen-like ions) and are characterized by three quantum numbers: n (the principal quantum number), l (the angular momentum quantum number), and m_l (the magnetic quantum number).

The specific form of the wavefunction depends on these quantum numbers, and there are mathematical expressions to calculate the radial and angular parts of the wavefunction for each allowed set of quantum numbers.

Given:

Lz = ℏ

Explanation:

According to Variation Theorem, eigen value for L_z is

L_z = mℏ

∴ m =1

⇒ Calculating Probability for m=1 ie. Y₁₁

\(P(m=1)={P(Y_{11}) \over Total Proliability}\)

\(P(m=1)= {(1/\sqrt{2})^2 \over (1/\sqrt{2})^2 + 1^2}\)

\(P(m=1)={1\over3}\)

Conclusion:

The probability for L_z that will give eigen value of h is \({1\over 3}\).